200 
- the 3 coordinates in object space (xp,yp,zp); 
- the interferometric phase d>u. 
For each GCP it is possible to write one range equation, 
one Doppler equation and one interferometric equation 
(see Figure 1). The 3 equations can be expressed as: 
+ (y , M - yp) 2 + ( z m -z p) 2 - Rq - AR • (col M -1) = 0 
Let (x,y) be the approximate values and (r|, £) the 
relative corrections of parameters and observables; we 
have: 
x = x +q 
y = y+S 
txpanaing to; io me Tirsi oraer, we get: 
>/(xJ-^Xp)^T(ÿs^-yp)^+(zs-^p^"- Rq - AR ■ (col M -1) 
4-71 
(xp -X M ) vX M + (y p -y M ) vy M + ( Z P - z m)‘ vz M 
+ i±ü..[R 0+A R.(col M -l)]=0 
where: 
xm, yM, zm are the master satellite coordinates 
vxm, vyM, vzm are the master velocity components 
x^ = 3g +a-| • t + a 2 -t 2 + a 3 • t 3 + 84 ■ t 4 + 35 • t 3 
yM =bo + bi t + b 2 -t 2 + b 3 -t 3 + b 4 -t 4 +b 5 t 5 
Z M =Co+C r t + c 2't 2 + c 3't 3 +C4't 4 +C5'| 5 
vx M — 83 + 2 a 2 t + 3 a 3 • t 2 + 4 • a 4 • t 3 + 5- 35 ■ t 4 
vy M = b, + 2■ b 2 • t + 3• b 3 • t 2 + 4• b 4 • t 3 + 5- b s • t 4 
vz^ = c, + 2 c 2 t + 3 c 3 • t 2 + 4 c 4 - t 3 + 5- C5 • t 4 
g(x,y)- 
{Sy j\x=x 
l x.x' 5+ [i7j|x.x'’ 1=0 
|y=y |y=y 
In matrix form, the linearised system becomes: 
A-ri+b = D-5 (9) 
where: 
A is the [(3 • ngcp + unk) , unk] matrix containing 
terms as |—I 
V$x J x=x 
|y=y 
q is the [unk , 1] vector that contains the parameter 
corrections, 
£, is the [(4-ngcp + unk) , l] vector that contains the 
observation corrections, 
D is the [(3-ngcp +unk), (4-ngcp + unk)] matrix 
containing terms: - — 
Uy J|x=x 
y=y 
Xs, ys, Zs 
are the slave satellite coordinates (also 
b 
is the [(3 • ngcp + unk) , l] vector containing terms 
described by polynomial of the 5 th order) 
as g(x,y), 
coIm 
is the slant range coordinate of the 
master image 
unk 
is the number of unknown parameters (14 in this 
case), 
fo_M 
is the Doppler frequency of the master 
image. 
ngcp 
is the number of GCPs. 
To the equations written for each GCP are added the so- 
called pseudo-observations of the unknown parameters: 
Pk = Pk ! w K 
where: 
Pk is the K th unknown parameter (to be adjusted), 
p K is the approximate value of the K th parameter p K 
(treated as pseudo-observation), 
w K is the weight associated to the K* 1 pseudo 
observation. 
The system of observations and pseudo-observations is 
neither linear with respect to the 14 unknown parameters 
(Ro, AR, To, AT, foo, foi, fD2, fD3, do, d-i, d2, d3, d 4 and ds) 
nor with respect to the observables (xp,yp,zp,Ou). Hence it 
is linearised and solved iteratively. The system can be 
written as: 
g(x, y) = 0 (8) 
where g is a non linear function, x represents the 
unknown parameters and y the observables. 
A.1 Stochastic Model and Solution 
To the observation and pseudo-observation vector is 
associated the variance-covariance matrix Cyy: 
|p GCP, ] 
0 
0 
0 
0 
0 
0 
[pGCFk J 0 
1 
0 
0 
0 
0 
0 
L C GCP|«j 
0 
0 
0 
0 
0 
0 
2 
0 Param-i 
0 
0 
0 
0 
0 
0 
2 
0 Pararri|_ 
0 
0 
0 
0 
0 
0 
2 
a ParamN. 
where: 
°xy Q xz 
0 
[ C GCP|J = 
a xy 
Oy Oy Z 
2 
0 
CT XZ 
CT yz CT Z 
0 
0 
0 0 
2 
0 <D